Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabautyʹs Theorem, which applies when the genus of a curve is greater than the rank of the Mordell–Weil group of the Jacobian. When Chabautyʹs Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabautyʹs Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabautyʹs Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate–Shafarevich group killed by a bielliptic isogeny.